| L(s) = 1 | + 3-s − 5-s + 3·9-s + 11-s + 10·13-s − 15-s + 17-s − 6·19-s + 4·23-s + 8·27-s + 6·29-s + 2·31-s + 33-s − 8·37-s + 10·39-s + 20·41-s − 4·43-s − 3·45-s − 7·47-s + 51-s + 2·53-s − 55-s − 6·57-s + 14·59-s − 8·61-s − 10·65-s − 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 9-s + 0.301·11-s + 2.77·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s + 0.834·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s − 1.31·37-s + 1.60·39-s + 3.12·41-s − 0.609·43-s − 0.447·45-s − 1.02·47-s + 0.140·51-s + 0.274·53-s − 0.134·55-s − 0.794·57-s + 1.82·59-s − 1.02·61-s − 1.24·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.353962477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.353962477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40511142129460231541305498386, −9.789927330029288171510856477396, −9.202053634679022763824735940935, −8.789272231445184092629170714700, −8.590140956839011974167373014814, −8.345606578897386903466460853075, −7.72279916913196250510551119877, −7.37178789505501947467188505608, −6.61989014358559293073065467617, −6.55081251014903475283625746215, −6.07071845619069716537708830873, −5.55757060813207654062647818198, −4.57731960000913253770588061262, −4.55342840324492373718308685332, −3.84901127216902779642437630539, −3.53332478968547246071695883085, −2.98478980778539671612450640472, −2.24843793034550670062404991118, −1.35421397992367284149744390398, −0.985516337670595522033698410727,
0.985516337670595522033698410727, 1.35421397992367284149744390398, 2.24843793034550670062404991118, 2.98478980778539671612450640472, 3.53332478968547246071695883085, 3.84901127216902779642437630539, 4.55342840324492373718308685332, 4.57731960000913253770588061262, 5.55757060813207654062647818198, 6.07071845619069716537708830873, 6.55081251014903475283625746215, 6.61989014358559293073065467617, 7.37178789505501947467188505608, 7.72279916913196250510551119877, 8.345606578897386903466460853075, 8.590140956839011974167373014814, 8.789272231445184092629170714700, 9.202053634679022763824735940935, 9.789927330029288171510856477396, 10.40511142129460231541305498386
